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Sequences for the coefficients of a few of the
first generating functions, in the case at hand:
1,1 1,1,1 1,1,2,1 1,1,2,2,1 1,1,2,3,3,1 1,1,2,3,4,3,1 1,1,2,3,5,5,4,1 1,1,2,3,5,6,7,4,1 1,1,2,3,5,7,9,8,5,1 1,1,2,3,5,7,10,11,10,5,1 1,1,2,3,5,7,11,13,15,12,6,1 1,1,2,3,5,7,11,14,18,18,14,6,1